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15.Math-Review

Monday 8/14/00Monday 8/14/00

15.Math-Review 2

General Mathematical Rules

( ) ( ), ,

0 , ( ) 0

a b c a b c a b b a

a a a a

Addition Basics:

1 21

n

i ni

x x x x

Summation Sign:

1

( 1) 1 2

2

n

i

n ni n

Famous Sum:

15.Math-Review 3

General Mathematical Rules

1 1

( ) ( ), ,

1 , if 0 ( ) 1a

ab c a bc ab ba

a a a a a a

Multiplication Basics

2 2 2

2 2 2

2 2

( ) 2 ,

( ) 2 ,

( )( )

a b a ab b

a b a ab b

a b a b a b

Squares:

3 3 2 1 1 2 3

3 3 2 1 1 2 3

( ) 3 3 ,

( ) 3 3

a b a a b a b b

a b a a b a b b

Cubes:

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Multiplication General Binomial Product:

1

( )n

n i n i

i

n

ia b a b

General Mathematical Rules

1 21

n

i ni

x x x x

Product Sign:

acabcba )( Distributive Property:

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Fractions Addition:

General Mathematical Rules

a b a b

c c c

a c ad bc

b d bd

ac

db

a b ab

c d cd a b a

b d d

Product:

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General Mathematical Rules

Powers Interpretation:

times

what if (0,1) ?? , a

aax xx x

0 1

1

1, ,

, ( ) , ( ) ,

1 1, ,

a b a b a a a a b ab

a aa a b

b

x x x

x x x x y xy x x

xx x x

x x x

General rules:

12

0

11

1

nni n

i

aa a a a

a

2

0

11 , if 1

1i

i

a a a aa

Series:

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General Mathematical Rules

Logarithms Interpretation:

The inverse of the power function. logxaa c x c

(where 2.71828...), log ln

log 1 0, log 1

log log

log

log log log

log log

e

b b

cb

c

b b b

nb b

ex x

b

aa

b

cd c d

c n c

General rules and notation:

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Exercises: We know that project X will give an expected yearly return of $20 M

for the next 10 years. What is the expected PV (Present Value) of project X if we use a discount factor of 5%?

How long until an investment that has a 6% yearly return yields at least a 20% return?

General Mathematical Rules

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Definition:

Graphical interpretation:

The Linear Equation

( )y x y ax c

c

1

a

-c/a

y

x

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Example: Assume you have $300. If each unit of stock in Disney Corporation costs $20, write an expression for the amount of money you have as a function of the number of stocks you buy. Graph this function.

Example: In 1984, 20 monkeys lived in Village Kwame. There were 10 coconut trees in the village at that time. Today, the village supports a community of 45 monkeys and 20 coconut trees. Find an expression (assume this to be linear) for, and graph the relationship between the number of monkeys and coconut trees.

The Linear Equation

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System of linear equations2x – 5y = 12 (1)

3x + 4y = 20 (2)

Things you can do to these equalities:(a) add (1) to (2) to get:

5x – y = 32

(b) subtract (1) from (2) to get:

x + 9y = 8

(c) multiply (1) by a factor, say, 4

8x – 20y = 48

All these operations generate relations that hold if (1) and (2) hold.

The Linear Equation

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Example: Find the pair (x,y) that satisfies the system of equations:2x – 5y = 12 (1)

3x + 4y = 20 (2)

Now graph the above two equations.

Example: Solve, algebraically and graphically,2x + 3y = 7

4x + 6y = 12

Example: Solve, algebraically and graphically, 5x + 2y = 10

20x + 8y = 40

The Linear Equation

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Exercise: A furniture manufacturer has exactly 260 pounds of plastic and 240 pounds of wood available each week for the production of two products: X and Y. Each unit of X produced requires 20 pounds of plastic and 15 pounds of wood. Each unit of Y requires 10 pounds of plastic and 12 pounds of wood. How many of each product should be produced each week to use exactly the available amount of plastic and wood?

The Linear Equation

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Definition:

Graphical interpretation:

The Quadratic Equation

2( )y x y ax bx c

y

x

When a<0

r2r1

y

x

When a>0

r2

c

r1

y

x

Can have only 1 or no root.

r1

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Completing squares:

The Quadratic Equation

2 2

2

2

2 2

2

4 4

2 4

b b b

a a a

b b

a a

y ax bx c a x x c

a x c

2)( hxaky Another form of the quadratic equation:

a

bck

a

bh

4 ,

2

2

The point (h,k) is at the vertex of the parabola. In this case:

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Example: Find the alternate form of the following quadratic equations, by completing squares, and their extreme point.

The Quadratic Equation

?483

?62

2

xx

xx

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Solving for the roots We want to find x such that ax2+bx+c=0. This can

be done by:Factoring.

Finding r1 and r2 such that ax2+bx+c = (x- r1)(x- r2)

The Quadratic Equation

0483

062

2

xx

xx Example:

a

acbbrr

2

4,

2

21

Formula

0483

062

2

xx

xx Example:

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Exercise: Knob C.O. makes door knobs. The company has estimated that their revenues as a function of the quantity produced follows the following expression:

The Quadratic Equation

5000510)( 2 qqqf where q represents thousands of knobs, and f (q), represents thousand of dollars.

If the operative costs for the company are 20M, what is the range in which the company has to operate? What is the operative level that will give the best return?

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Definition: For 2 sets, the domain and the range, a function associates for

every element of the domain exactly one element of the range. Examples:

Given a box of apples, if for every apple we obtain its weight we have a function. This maps the set of apples into the real numbers.

Domain=range=all real numbers.

For every x, we get f(x)=5.

For every x, we get f(x)=3x-2.

For every x, we get f(x)=3 x +sin(3x)

Functions

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Types of functions Linear functions Quadratic functions Exponential functions: f(x) = ax

Example: Graph f(x) = 2x , and f(x) = 1-2-x.

Example: I have put my life savings of $25 into a 10-year CD with a continuously compounded rate of 5% per year. Note that my wealth after t years is given by w = 25e5t. Graph this expression to get an idea how my money grows.

Functions

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Types of functions Logarithmic functions

f(x) = log(x) Lets finally see what this ‘log’ function looks like:

Functions

-8

-6

-4

-2

0

2

4

6

8

-8 -3 2 7

f(x)=exp(x)

f(x)=ln(x)

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Given a function f(x), a line passing through f(a) and f(b) is given by:

Convexity and Concavity

number. real a ),()1()()( bfafyy

.]1,0[ ),)1(()()1()( bafbfaf

.]1,0[ ),)1(()()1()( bafbfaf

Definition: f(x) is convex in the interval [a,b] if

f(x) is concave in the interval [a,b] if

Another definition is f(x) is concave if -f(x) is convex

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These ideas graphically:

Convexity and Concavity

y

xba

f(a)

f(b)

)())()((

)()1()(

bfbfaf

bfafy

xba

f(a)

f(b)

)()1()( bfaf

))1(( baf

1